Junkao Mathematics - independent shared foundations key terms

A function expressible as one polynomial divided by another, defined wherever the denominator is nonzero. Its graph may have vertical asymptotes at denominator zeros and a horizontal or oblique asymptote determined by degrees.

A decomposition of a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator. It makes integration and series expansion of rational functions tractable.

The statement that a polynomial f has factor (x - a) exactly when f(a) = 0. It converts root-finding into factorisation and underpins the solution of polynomial equations and inequalities.

The result that dividing a polynomial f(x) by (x - a) leaves remainder f(a). It allows remainders to be computed by substitution instead of long division, and generalises to divisors of higher degree.

An expression formed from a variable and constants using only addition, subtraction and non-negative whole-number powers. Its degree is the highest power present, and it is defined and continuous for all real inputs.

The function giving the non-negative distance of a number from zero, written |x| and defined as x when x is non-negative and -x otherwise. Its graph is V-shaped and equations involving it often split into cases.

A line that a curve approaches ever more closely as the variable tends to infinity or to a value where the function is undefined. Vertical, horizontal and oblique asymptotes are all examined for rational and exponential graphs.

A sequence in which consecutive terms differ by a fixed common difference. Its nth term is a + (n-1)d and the sum of the first n terms is n/2 times the first term plus the last term.

A sequence in which each term is a fixed multiple, the common ratio, of the previous one. Sums converge only when the ratio lies strictly between -1 and 1, giving the sum to infinity a/(1-r).

The property of a series whose partial sums approach a finite limiting value as more terms are added. For a geometric series this happens exactly when the common ratio has modulus less than one.

Compact notation using the Greek capital sigma to denote a sum, with an index, lower and upper limits written beneath and above. It is used to state series sums and to manipulate standard results algebraically.

A summation technique in which each term of a series is written as a difference of consecutive values of some expression, so most terms cancel when the sum is expanded, leaving only a few boundary terms.

An equation defining each term of a sequence using one or more earlier terms together with an initial value. Solving it means finding a closed formula, often verified afterwards by induction.

The expression of a power of a sum, such as (a + b)^n, as a sum of terms involving binomial coefficients. For non-integer or negative powers it becomes an infinite series valid over a restricted range.

An angular measure in which one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn is 2 pi radians, and calculus formulas for trigonometric functions require this measure.

The distance along a section of a curve. For a circle of radius r subtending angle theta in radians it equals r times theta, and for a general curve it is found by integrating the square root of one plus the squared derivative.

An equation true for all values of the variables, such as the Pythagorean identity linking sine and cosine, or double-angle formulas. Identities are used to simplify expressions and to transform integrals.

The replacement of sine of a small angle by the angle itself in radians, with cosine approximated by one minus half the angle squared. These approximations underpin the derivative of sine and simple harmonic motion.

The value a function or sequence approaches as its input approaches a specified point or grows without bound. Limits define continuity, derivatives and convergence, and can be evaluated by algebra or numerically.

The instantaneous rate of change of a function, defined as the limit of a difference quotient. Geometrically it is the gradient of the tangent, and it measures sensitivity of the output to small changes in input.

The rule for differentiating a composite function: differentiate the outer function at the inner function and multiply by the derivative of the inner function. It is a central technique in this independent calculus practice.

A technique for differentiating a relation not given in the form y equals f(x), by differentiating every term with respect to x and treating y as a function of x. It produces derivatives involving both variables.

A point where the derivative of a function is zero, so the tangent is horizontal. It may be a maximum, a minimum or a point of inflection, distinguished by the second derivative or a sign change.

The derivative of the derivative, measuring how the gradient itself changes. Its sign at a stationary point usually classifies it as a maximum when negative or a minimum when positive.

A vector giving the location of a point relative to a chosen origin, written as a column or in unit-vector notation. Differences of position vectors give displacement vectors between points.

The product of two vectors giving a number: the product of their magnitudes and the cosine of the angle between them. It tests perpendicularity, since it is zero exactly when the vectors are perpendicular.

A description of a line as a fixed position vector plus a scalar multiple of a direction vector. It encodes every point on the line as the parameter varies over all real values.

Selections of k elements from n without regard to order, counted by C(n,k) = n!/(k!(n−k)!); used when only the group matters.

Ordered arrangements of all or part of a set of elements; the number of orderings of n distinct items is n!, and repeated items require division by their factorials.

Probability of A given that B has occurred, P(A|B) = P(A∩B)/P(B), defined when B has non-zero probability.